Belyi's theorem

Belyi's theorem states that a smooth projective curve over the complex numbers is definable over the algebraic numbers exactly when it admits a finite map branched only over 0, 1, and infinity.

Belyi map: A finite morphism from a smooth projective curve to the projective line, branched only over 0, 1, and infinity. Its existence is the geometric condition that characterizes curves defined over the algebraic numbers.
Algebraic number: A complex number that is a root of a nonzero polynomial with rational coefficients. The theorem identifies definability over the algebraic numbers as the arithmetic side of its criterion.
Gennadi Belyi: A Soviet and Russian mathematician known for proving the theorem that bears his name. His 1979 result established the characterization of curves defined over the algebraic numbers.
Monodromy group: The permutation group describing how sheets of a covering are permuted by loops around branch values. The monodromy of a Belyi map can be computed from its branch cycles.
Riemann existence theorem: A theorem identifying finite branched covers of the complex projective line with suitable topological covering data. It constructs covers from topological data, while Belyi's theorem imposes an arithmetic criterion on curves.
Branched covering: A map that is locally a covering except at points where its local behavior has ramification. Belyi maps are branched coverings with just three permitted branch values.
Algebraic curve: A one-dimensional algebraic variety, possibly with singularities or multiple components. The theorem concerns the smooth projective curves within this broader class.
Alexander Grothendieck: A mathematician whose work reshaped algebraic geometry and introduced the theory of dessins d'enfants. He recognized the combinatorial and Galois-theoretic significance of Belyi's theorem.
Branch cycle description: A description of a branched cover by permutations associated with loops around its branch values. For a Belyi map, three branch cycles encode the covering and obey a product relation.
Hurwitz space: A moduli space parametrizing branched covers with specified degree and branching data. Belyi maps form special covers whose branch values are fixed to three points.

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